File:Chebyshev-big.svg

In mathematics, in the field of homological algebra, given an abelian category ${\displaystyle \mathcal{𝒞}}$ having enough injectives and an additive (covariant) functor

${\displaystyle F:\mathcal{𝒞}\to \mathcal{𝒟}}$,

an acyclic object with respect to ${\displaystyle F}$, or simply an ${\displaystyle F}$-acyclic object, is an object ${\displaystyle A}$ in ${\displaystyle \mathcal{𝒞}}$ such that

${\displaystyle {\mathrm{R}}^{i}F(A)=0}$ for all ${\displaystyle i>0}$,

where ${\displaystyle {\mathrm{R}}^{i}F}$ are the right derived functors of ${\displaystyle F}$.

References

• S. Caenepeel, Brauer Groups, Hopf Algebras, and Galois Theory, Springer Verlag, 1998, ISBN 0-7923-4829-X. P.454.